Solving a system of multivariable equations

In summary: Now, the equation $F(x,y,z) = 0$ has the unique solution $G(z) = xe^y + yf(z)$ for all $z \in U$. Therefore, $x$ and $y$ are differentiable functions of $z$ locally, and the original system can be solved.In summary, we found that the equation system given in the original post can be solved for $x$ and $y$ locally provided that we assume that the derivatives of $f$ and $g$ are continuous.
  • #1
maxjohnson
2
0
Consider the equation system

x*e^y + y*f(z) = a
x*g(x,y) +z^2 = b

where f(z) and g(x,y) are differentiable functions, and a and b are constants. Suppose that the system defines x and y as differentiable functions of z. Find expressions for dx/dz and dy/dz.

Any help would be appreciated!

Thank you!
 
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  • #2
Hello maxjohnson and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 
  • #3
This is what I’ve got so far:

1. dxe^y + dyxe^y + dyf(z) + yf’(z)dz = 0

2. g(x,y) + x(dg(x,y)/dx)dx + x(dg(x,y)/dy)dy = -2zdz
 
  • #4
maxjohnson said:
This is what I’ve got so far:

1. dxe^y + dyxe^y + dyf(z) + yf’(z)dz = 0

2. g(x,y) + x(dg(x,y)/dx)dx + x(dg(x,y)/dy)dy = -2zdz

Hi maxjohnson! Welcome to MHB! (Smile)

Good.
Now divide both equations by dz, so that they will contain $\d x z$ and $\d y z$.
That means we have 2 equations with $\d x z$ and $\d y z$ as unknowns, which we can solve can't we? (Wondering)
(Click Reply With Quote if you want to know how to format derivatives like that.)
 
  • #5
I'm not so fond of treating derivatives as quotients of differentials, but I think formally it works out here.

Without wanting to be argumentative, I would like to add some comments that I hope could be helpful for the OP and/or others. The original system from post #1 is
\[
\left\{
\begin{aligned}
x e^y + y f(z) &= a\\
x g(x,y) +z^2 &= b
\end{aligned}
\right.
\]
It was given that the system defines $x$ and $y$ as differentiable functions of $z$, but how do we find out whether this is indeed the case? This can be done together with the rest of the problem along the following lines, at least locally and provided that we are allowed to assume additionally that the derivatives of $f$ and $g$ are continuous.

The above system can be written as $F(x,y,z) = 0$ with $F : \mathbb{R}^2 \times \mathbb{R} \to \mathbb{R}^2$ given by
\[
F(x, y, z) =
\begin{pmatrix}
x e^y + y f(z) - a\\
x g(x,y) +z^2 - b
\end{pmatrix}
\]
Let $D_1F$ be the partial derivative of $F$ with respect to its first two arguments. At any point $(x_0,y_0,z_0) \in \mathbb{R}^3$ where $F(x_0,y_0,z_0) = 0$ and the $2 \times 2$ matrix $D_1F(x_0,y_0,z_0)$ is non-singular, the Implicit Function Theorem applies: There exists a continuously differentiable function $G$, defined in a neighborhood $U$ of $z_0$ and taking values in a neigborhood $V$ of $(x_0,y_0)$, such that for each $z \in U$ the equation $F(x,y,z) = 0$ has the unique solution $G(z) \in V$. We may differentiate the relation $F(G(z),z) = 0$ with respect to $z \in U$ to find
\[
D_1F(G(z),z)DG(z) + D_2F(G(z),z) = 0,
\]
so $DG(z) = -[D_1F(G(z),z)]^{-1}D_2F(G(z),z)$ for all $z \in U$. (The non-singularity of $D_1F(G(z_0),z_0)$ and the continuity of the derivatives of $f$ and $g$ are used together to ensure that the matrix inverse exists for $z$ near $z_0$.)
 

FAQ: Solving a system of multivariable equations

1. How many equations and variables are needed to solve a system of multivariable equations?

A system of multivariable equations typically requires the same number of equations as the number of variables. For example, a system with three variables would need three equations to solve it.

2. What is the best method for solving a system of multivariable equations?

The best method for solving a system of multivariable equations depends on the specific equations and variables involved. Some common methods include substitution, elimination, and graphing. It is important to choose a method that is most efficient for the given system.

3. Can a system of multivariable equations have more than one solution?

Yes, a system of multivariable equations can have zero, one, or infinitely many solutions. This depends on the specific equations and variables involved. It is important to check the solution(s) to ensure they satisfy all of the equations in the system.

4. How do I know if a system of multivariable equations is consistent or inconsistent?

A consistent system of multivariable equations has at least one solution that satisfies all of the equations, while an inconsistent system has no solutions that satisfy all of the equations. This can be determined by solving the system and checking if the solution(s) satisfy all of the equations.

5. Are there any shortcuts or tricks for solving a system of multivariable equations?

While there are no shortcuts or tricks that apply to every system of multivariable equations, there are some techniques that can make the process easier. These include using matrices and row operations, using symmetry to simplify equations, and using substitution or elimination strategically. It is important to practice and familiarize yourself with various methods to become more efficient at solving these types of equations.

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